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Peter Sells (1995) calls ないで a “verbal gerund” and なくて an “adjectival gerund”. When you have participial constructions, it does not make a difference which one you use, but Sells notices that only the verbal gerund can be selected by an auxiliary verb:

食べないでおいた
* 食べなくておいた (Sells 1995:287)

Similarly, when you want to use these forms adverbially as in your examples, you have to use ないで. You cannot replace it with なくて. If you do so, they will be interpreted as participial constructions.

何もしないで暖かくする (ambiguous)
'not do any activity, and keep the body warm'
'keep the body warm by not doing any activity'

何もしなくて暖かくする (unambiguous)
'not do any activity, and keep the body warm'
* 'keep the body warm by not doing any activity'